Evaporation Timescales of HI Clouds

Masahiro Nagashima (Nagasaki)with

Hiroshi Koyama (Kobe), Shu-ichiro Inutsuka (Kyoto)

1. Introduction2. 1D Simulation of cloud evaporation3. Analytic Approximation Forumula4. Summary

ref) Nagashima, Koyama & Inutsuka, 2005, MNRAS, 361, L25 (astro-ph/0503137) Nagashima, Inutsuka & Koyama, submitted (astro-ph/0603259)

2006/02/06 2

Discovery of Tiny HI clouds

sub-pc (~0.01pc) clouds“tiny” HI clouds recently discovered!

Braun & Kanekar (2005)Stanimirovic & Heiles (2005)

simple interpretation: CNM cloud surrounded by WNM

Growth rate? ... Most fundamental quantity

N~1018 /cm2

2006/02/06 3

Formation of Tiny HI Cloudsthermal instability and turbulance behind shock fronts: many clumps form

... one of natural mechanisms of HI clouds formation

Once the clouds formed,how they evolve?

Koyama & Inutuka (2002)

Myers (1978)

Log[n/cc]

Log[T/K]

constantpressure

Physical State of Diffuse ISM Clouds

P /k~103

diffuse cloudsWNM/CNM

under pressure equilibrium

Physical State of Diffuse Clouds

Diffuse (n<102cc) gas in the ISM:

- Co-existence with two phases　Warm Neutral Medium (WNM) 　Cold Neutral Medium (CNM) - pressure balance with heat-conductive interface- negligible self-gravity

~ 1G

~1.2×108 ncm−3

−1/2

yr

l~2×103 T102K

1/2

ncm−3

−1/2

pc

~ Galactic rotation

~ Galaxy size

T~104K , n~10−1/cm3

T~102K , n~101/cm3

x

T

CNM

WNM

2006/02/06 6

Heating/Cooling rates

Koyama & Inutsuka (2000)

Our results do NOT depend on the detailsof the heating/cooling processes

2006/02/06 7

Thermal instability

sequence of heating (Γ)= cooling (Λ)

stable eq.

unstable eq.Γ＜Λ

Γ＞Λ

WNMCNM

phase transition between WNM and CNM

Koyama & Inutsuka's cooling function used

Frontal motion between WNMand CNM corresponds toCNM domain (cloud) growth

at “saturation pressure” p_sat,Γ＝Λ over the whole system

n

P/k

T

∵pressure constant

2006/02/06 8

Evaporation & Condensation

whole system cooling ：condensation

whole system heating ：evaporation

n

p

cooling > heating

cooling < heating

static front(in 1D)

evaporation

condensation p psat

p psat

→saturation pressure

p psat

p psat

Pressure determines whether evaporation or condensation

Describing evaporation and condensation of clouds

2006/02/06 9

Basic equations

∂∂ t

∇⋅v=0

∂ v∂ t

v⋅∇v=−1∇ p

∂e∂ t

∇⋅e v p∇⋅v=−n2−n∇⋅∇ T

e= p /−1p=n k BT ∝T 1/2

Lheat-loss function

conductivity for neutral gas

L

0 Tlike a cubic function

continuity

EOM

energy eq.

2006/02/06 10

Characteristic scales

　

cool≃kTn

≃104−106 yr≃3×1011−13 s

cS≃105 T102K

1/2

cm/ s

lF= Tn2

≃3×1015−1018cm

cooling time

sound velocity

Field length: cooling = thermal conductionproviding the thickness offront (transition layer)

vl F /cool≃104cm / s≪cS

ISM is quickly separated into two phasesthen the domain grows slowly

2006/02/06 11

Numerical Simulation

Numrical simulation for spherically symmetric clouds

1D (radial) 、 Lagrangean mesh (2000), 2nd-order Godunovboundary condition: a constant outside pressure (p_WNM)

example (p=p_sat=2823kB; saturation pressure)

Radius of clouds isdefined at which thedensity is half the CNM density

cloud center WNM

snapshots ofa cloud evaporating

2006/02/06 12

Evaporation rate dM/dt

Full Numerical Simulation

Quasi-steady state(QSS) numerical solution

dM/dt at surface

McKee&Cowie's dM/dt

- Numerical simulation is consistent with quasi-steady state ev.- McKee-Cowie's dM/dt overestimates by a factor of 4-5

dM/dt(r)

Case of p=p_sat

Log[r/pc]

dM/dt

dM r dt

=4 r2r v r

2006/02/06 13

Evaporation Timescale

R~0.01pc clouds evaporate in ~Myr

If tiny HI clouds exist ubiquitously, some mechanisms are neededto create the clouds continuously.

expectation from the analytic formula

condensation

Critical Size for Condensation

- When , static clouds can exist.- Static clouds are always unstable.- 0.01pc clouds must evaporate in the standard case of Γ.

p psat

2006/02/06 15

Analytic Approximation Formula

1. Isobaric approximation because of much slower motion than sound velocity Tds=dU+pdV =d(U+pV)-Vdp =dH-Vdp→Tds=dH

2. Assuming quasi-steady state (QSS)

∂∂ t

−R ∂∂r

≡−R∂r

R : speed of a front

R

R0v>0

(in case of evaporation）

−1

k B

dTdt

− dpdt

=−L ∇⋅∇ T

−1

k B

dTdt

=−L ∇⋅∇ T

2006/02/06 16

−1

k B

ud ∂rT=−L ∂r∂rTd−1r

∂rT

Analytic Approximation Formula

in case of d-dim. spherically symmetric, energy eq. becomes

replace with LHS for d=1

−1

k B

u1∂rT=−L ∂r∂rTassuming the same

−1

k B

ud ∂rT=

−1k B

u1∂rTd−1r

∂rT

ud ∂rT=u1∂rT−1

k Bd−1r

∂rTor,

2006/02/06 17

ud ∂rT=u1∂rT−1

k Bd−1r

∂rT

Analytic Approximation Formula

・non-zero dT/dr only at front ・substitution LHS in d=1 for 1st and 2nd terms in RHS

non-zero only at r=R (cloud surface)

(fluid velocity) ＝ (fluid velocity for d=1) + (curvature term)

a non-trivial solution is obtained at r=R, so

ud R=u1R−1

k Bd−1R

RR

mean curvature

2006/02/06 18

Analytic Approximation Formula

Curvature term is very similar to McKee & Cowie (1977)'s evaporation ratewithin a factor of 4-5

M=−4CNM R2 R

∝R∝M 1/3 for R≪Rcrit

By further transformation,

Evaporation rate:

≡V p 1− RcritR ∝R2∝M 2 /3 for R≫Rcrit

substantially differentfrom MC77 for・large p_WNM・large clouds

u R

R=V p−−1

k Bd−1R

RCNM

M=16RR

5 k B

2006/02/06 19

Evaporation Timescale

R~0.01pc clouds evaporate in ~Myr

expectation from the analytic formula

condensation

2006/02/06 20

Summary- Clouds of 0.01 pc evaporate in 1Myr- for >0.1 pc depends strongly on -> large clouds can grow

- TIny clouds must be formed continuously, or, ambient pressure would be much higher than

standard value, probably because of higher heating rate.

- Study of statistical properties such as mass function is important for future analysis

evap PWNM

P /k B≃103.5